How to solve this trignometric equation for all values between 0 <= x <= 2pi. The equation is sin^2(x) - 1 = 0

To solve the trigonometric equation \( \sin^2(x) - 1 = 0 \) for \( 0 \leq x \leq 2\pi \), follow these steps:

1. **Rewrite the Equation:**

   The given equation is:
   \[
   \sin^2(x) - 1 = 0
   \]

   Add \( 1 \) to both sides to isolate \( \sin^2(x) \):
   \[
   \sin^2(x) = 1
   \]

2. **Solve for \( \sin(x) \):**

   To find the values of \( x \), take the square root of both sides:
   \[
   \sin(x) = \pm 1
   \]

3. **Find the Angles Where \( \sin(x) = 1 \) and \( \sin(x) = -1 \):**

   - For \( \sin(x) = 1 \), this occurs when \( x = \frac{\pi}{2} \).
   - For \( \sin(x) = -1 \), this occurs when \( x = \frac{3\pi}{2} \).

4. **List the Solutions:**

   Therefore, the values of \( x \) that satisfy the equation within the interval \( 0 \leq x \leq 2\pi \) are:
   \[
   x = \frac{\pi}{2}, \quad x = \frac{3\pi}{2}
   \]

So, the solution to the equation \( \sin^2(x) - 1 = 0 \) within the interval \( 0 \leq x \leq 2\pi \) is:

\[
x = \frac{\pi}{2}, \quad \frac{3\pi}{2}
\]

Would you like more details or have any questions about this solution?

### Related Questions:
1. How would you solve \( \cos^2(x) - 1 = 0 \) for \( 0 \leq x \leq 2\pi \)?
2. What are the general solutions to \( \sin^2(x) = 1 \) over all real numbers?
3. How do you find solutions to \( \sin(x) = 0 \) within the interval \( 0 \leq x \leq 2\pi \)?
4. Can you solve \( \sin(2x) = 1 \) for \( 0 \leq x \leq 2\pi \)?
5. What are the solutions to \( \cos(x) = 1 \) for \( 0 \leq x \leq 2\pi \)?

### Tip:
To solve trigonometric equations, it's helpful to recall the unit circle values where sine and cosine have specific known values, like \( 0, \pm 1, \pm \frac{\sqrt{2}}{2} \).

Learn how to solve the trigonometric equation sin^2(x) - 1 = 0 for 0 <= x <= 2pi with a detailed step-by-step solution. This problem involves applying trigonometric identities to find specific angles where the equation holds true. Suitable for high school students studying trigonometry.

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